Question

Explain how can x1/2 + 1/3 is equivalent to the expression x1/3 x x1/2?

Answer 1

To demonstrate that the expressions x^(1/2) + 1/3 and x^(1/3) * x * x^(1/2) are equivalent, we can start by simplifying each expression individually.

1. Expression: x^(1/2) + 1/3
To add these terms, we need a common denominator. The denominator of 1/3 is already 3, so we'll convert x^(1/2) to have a denominator of 3 as well:
x^(1/2) can be expressed as x^(3/6) since multiplying the numerator and denominator by 3 gives us: (x^(1/2) * 3) / (2 * 3) = (3x^(1/2)) / 6
Now, our expression looks like this: (3x^(1/2))/6 + 1/3
Since 6 is now the common denominator, we can combine the terms: (3x^(1/2) + 2) / 6

2. Expression: x^(1/3) * x * x^(1/2)
We can multiply x^(1/3) and x using the properties of exponents, which states that when multiplying with the same base, the exponents are added together.
Therefore, x^(1/3) * x = x^((1/3) + 1) = x^(4/3)
Now, our expression becomes x^(4/3) * x^(1/2)
Again, using the exponent property, we can add the exponents of x: x^((4/3) + (1/2)) = x^(11/6)

So, the expressions x^(1/2) + 1/3 and x^(1/3) * x * x^(1/2) are not equivalent. The simplified forms of these expressions are (3x^(1/2) + 2) / 6 and x^(11/6), respectively.